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Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms

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  • El Alaoui, Ahmed

Abstract

We show that in the Ising pure p-spin model of spin glasses, shattering takes place at all inverse temperatures β∈((2logp)/p,2log2) when p is sufficiently large as a function of β. Of special interest is the lower boundary of this interval which matches the large p asymptotics of the inverse temperature marking the hypothetical dynamical transition predicted in statistical physics. We show this as a consequence of a ‘soft’ version of the overlap gap property which asserts the existence of a distance gap of points of typical energy from a typical sample from the Gibbs measure. We further show that this latter property implies that stable algorithms seeking to return a point of at least typical energy are confined to an exponentially rare subset of that super-level set, provided that their success probability is not vanishingly small.

Suggested Citation

  • El Alaoui, Ahmed, 2026. "Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms," Stochastic Processes and their Applications, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:spapps:v:192:y:2026:i:c:s0304414925002364
    DOI: 10.1016/j.spa.2025.104792
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